A deterministic framework for epidemic dynamics with abrupt intervention: Indicator-based switching and phase-dependent thresholds

Abstract

Mathematical models of epidemiology frequently assume time-invariant parameters, failing to capture the abrupt systemic shifts triggered by delayed medical interventions. In this study, we formulate a deterministic Susceptible-Infectious-Recovered (SIR) framework incorporating an abrupt, time-triggered intervention mechanism, modulated by a time-dependent indicator function, 1A(t), to explicitly simulate the sudden formulation and deployment of an antidote at a critical time, t1. This approach partitions the epidemic timeline, yielding a phase-dependent basic reproduction number (R0). The intervention time t1 determines when the system transitions from the pre-intervention reproduction regime to the post-intervention regime. We rigorously establish the positivity and boundedness of the system and derive the phasedependent disease-free and endemic equilibria. Utilizing the Next Generation Matrix and the Routh-Hurwitz criterion, we prove the local and global asymptotic stability of these states, identifying that the post-antidote autonomous subsystem undergoes a forward transcritical bifurcation at R^post_0 = 1. To translate these analytical theorems into mathematical intervention insights, we conduct comprehensive numerical and sensitivity analyses. Phase portraits and bifurcation diagrams illustrate the resulting global stability regimes and threshold boundaries. Furthermore, a two-dimensional synergistic parameter analysis of therapeutic curing and prophylactic vaccination demonstrates the nonlinear trade-offs in resource allocation. Crucially, by tracking cumulative disease-induced mortality density and varying the intervention delay (t1), we mathematically quantify the nonlinear cost of delayed intervention. Our findings demonstrate that the synergistic and timely deployment of an antidote is critical for controlling highly infectious viral diseases and preventing persistent endemic transmission.